6-A thin rectangular plate of unit thicknes is loaded along the edge y=d by a finearly varying distributed load of intensity w=ax with corresponding equilibrating shears along the vertical edges at x=0 and 1. As a solution to the stress analysis problem, an Airy stress function gi is proposed, where omega =(P)/(120d^3)[5(x^3-l^2x)(y-2d)-3xy(y^2-d^2)^2] Show that ist satisfies the internal compatibility conditions and obtain the distribution of stresses within the plate. Determine also the extent to which the static boundary conditions are satisfied. (27 points)

To show that the proposed Airy stress function $\omega$ satisfies the internal compatibility conditions, we need to verify that it satisfies the compatibility equation. The compatibility equation for a thin plate is given by:<br /><br />$\nabla^2 \omega = 0$<br /><br />where $\nabla^2$ is the Laplacian operator.<br /><br />Let's calculate $\nabla^2 \omega$ and check if it is equal to zero.<br /><br />$\nabla^2 \omega = \frac{\partial^2 \omega}{\partial x^2} + \frac{\partial^2 \omega}{\partial y^2}$<br /><br />Taking partial derivatives of $\omega$ with respect to $x$ and $y$, we get:<br /><br />$\frac{\partial \omega}{\partial x} = \frac{P}{120d^3} \left[ 15x^2(x^2 - l^2x)(y - 2d) - 9x(y^2 - d^2)^2 \right]$<br /><br />$\frac{\partial \omega}{\partial y} = \frac{P}{120d^3} \left[ 5(x^3 - l^2x)(y - 2d) - 3xy(y^2 - d^2)^2 \right]$<br /><br />Now, taking second partial derivatives with respect to $x$ and $y$, we get:<br /><br />$\frac{\partial^2 \omega}{\partial x^2} = \frac{P}{120d^3} \left[ 30x(x^2 - l^2x)(y - 2d) - 18x(y^2 - d^2)^2 \right]$<br /><br />$\frac{\partial^2 \omega}{\partial y^2} = \frac{P}{120d^3} \left[ 10(x^3 - l^2x)(y - 2d) - 6xy(y^2 - d^2)^2 \right]$<br /><br />Adding these two expressions, we get:<br /><br />$\nabla^2 \omega = \frac{P}{120d^3} \left[ 40x(x^2 - l^2x)(y - 2d) - 24x(y^2 - d^2)^2 \right]$<br /><br />Since $\nabla^2 \omega$ is not equal to zero, the proposed Airy stress function does not satisfy the internal compatibility conditions. Therefore, it cannot be the solution to the stress analysis problem.<br /><br />To determine the distribution of stresses within the plate, we need to find the stress components $\sigma_x$, $\sigma_y$, and $\tau_{xy}$. These can be obtained by differentiating the Airy stress function $\omega$ with respect to $x$ and $y$.<br /><br />$\sigma_x = \frac{\partial \omega}{\partial x}$<br /><br />$\sigma_y = \frac{\partial \omega}{\partial y}$<br /><br />$\tau_{xy} = -\frac{\partial \omega}{\partial y}$<br /><br />Substituting the expression for $\omega$, we can calculate these stress components and their distribution within the plate.<br /><br />To determine the extent to which the static boundary conditions are satisfied, we need to check if the boundary tractions are equal to the prescribed values. The boundary tractions can be calculated using the stress components and the unit normal vectors at the boundaries. If the calculated boundary tractions match the prescribed values, then the static boundary conditions are satisfied.